As is known in the optical communications and optical sensing arts, spatial modes are mathematical functions that can be used to describe transverse (i.e., perpendicular to direction of propagation of a light beam) spatial dependence of the complex amplitude of the electric field of a light beam. These mathematical functions are solutions to an electromagnetic wave equation. For example, the light beam of a conventional laser pointer is a spatial mode that is referred to as the fundamental spatial mode, i.e., the lowest order solution to the wave equation.
The spatial dependence of the intensity of the complex amplitude of the electric field of the fundamental spatial mode is characterized by being brightest at a light beam's center and, becoming gradually less bright farther from the beam's center. There are, however, high-order solutions to the wave equation, i.e., higher-order spatial modes. The complex electric field amplitudes of higher-order spatial modes have more complex spatial dependencies. For example, there are higher-order spatial modes referred to as Hermite-Gaussian modes (FIG. 2—top), and Laguerre-Gaussian modes (FIG. 2—bottom), which have ring-like and lobe-like spatial dependencies, respectively. Other higher-order spatial modes are Linearly Polarized modes and vector modes.
Higher-order spatial modes can propagate through free space (e.g. Earth's atmosphere, outer space) and waveguides (e.g. optical fibers). As a result, higher-order spatial modes are receiving significant interest in the communications and sensing arts.
For example, higher-order spatial modes can be used to increase the data speeds of free space and optical fiber communication at a given wavelength (i.e., spectral efficiency), where each higher-order spatial mode is used as a data state with which to encode data or, a channel over which data is encoded otherwise. Also, higher-order spatial modes can be used to enhance image resolution in microscopy, where image resolutions beyond wavelength dependent limits can be achieved via illumination with higher-order spatial modes (i.e., super-resolution).
For any spatial mode, its classification is often necessary, especially with respect to the applications noted above. For fundamental spatial modes, classification comprises characterization of the spatial modes' quality via the so-called M2 factor, i.e., a product of the beams' measured size and divergence. However, higher-order spatial modes are more various and, the complex amplitude of the electric field of each has more complex spatial dependence. Therefore, classification of higher-order spatial modes requires a more complex spatial analysis including, differentiation of the high-order spatial modes from each other. Measurement of the M2 factor is insufficient.
Canonical systems and methods to classify higher-order spatial modes comprise indirect measurement of the complex amplitude of a light beam's electric field. Typically, the complex amplitude of a light beam is indirectly measured using interferometry or, holographic techniques via unconventional optical devices/elements. Such unconventional optical devices/elements must emulate the complex spatial dependencies of the complex amplitudes of the electric fields of higher-order spatial modes. Unconventional optical devices/elements include, liquid crystal or micro-mirror based spatial light modulators or, custom reflective, refractive or, diffractive optical elements (e.g. spiral phase plates, q-plate, fork diffraction grating, meta-material).
While arguably effective, interferometry or, complex holographic techniques via unconventional optical devices/elements may have prohibitive complexity, i.e., dependence on a light beam's alignment, size, wave front (e.g. curvature, etc.), polarization, and wavelength. Additionally, unconventional optical devices/elements may have prohibitive cost and efficacy. They require quality of fabrication that depends on how well the complex spatial dependencies of the complex amplitudes of the electric fields of higher-order spatial modes can be emulated.